Summary and Analysis

Linear Momentum: Collisions

Terms and Formulae

Collisions in One Dimension, page 2

page 1 of 2

The most simple case of a collision is a one-dimensional, or head-on
collision. Because of the conservation of energy and momentum we are able to
predict a great deal about these collisions, and to calculate relevant
quantities after the collision occurs. Before we do so, however, we must define
exactly what is meant by a collision.

What is a Collision?

We all know, somewhat intuitively, the common meaning of a collision: two things
hitting each other. Whether the objects are two billiard balls, two particles,
or two cars, this common definition applies. The definition used in physics,
however, is something more precise. In physics, a collision has two aspects:

Two particles hit each other

A large force is felt by each particle for a relatively short amount of
time.

In this way, a collision can be seen as an event that imparts a large amount of
impulse to the
objects involved. In
addtion, recall that impulse changes momentum.

A typical collision problem involves two particles with known initial velocities
colliding; we are required to calculate the final velocity of each object. If
we knew the forces acting during the collision this would be easy. Usually,
however, we do not, and are forced to look for other methods of solving the
problem. For instance, two balls of the same masses and initial velocities upon
hitting a wall bounce back with different velocities according to the
"bounciness" or elasticity of the ball. We will examine the cases in which
collision problems are soluble, and make some general statements about
collisions.

Elastic Collisions

A special category of collisions is called elastic collisions.
Formally,
an elastic condition is one in which kinetic energy is conserved. This may be
difficult to grasp conceptually, so consider the following test: drop a ball
from a certain height. If it hits the floor and returns to its original height,
the collision between the ball and the floor is elastic. Otherwise it is
inelastic. Collisions between pool balls are generally elastic; car crashes are
generally inelastic.

Why are these collisions special? We know with all collisions that
momentum is conserved. If two particles collide we can use the following
equation:

m1v1o + m2v2o = m1v1f + m2v2f

However, we also know that, because the collision is elastic, kinetic energy is
conserved. For the same situation we can use the following equation:

m1v1o2 + m2v2o2 = m1v1f2 + m2v2f2

Again, we are usually given the masses and the initial velocities of the two
colliding particles, so we are given
m1
,
m2
,
v1o
and
v2o
. If we
use these equations together, we now have two
equations and two unknowns:
v1f
and
v2f
. Such a situation is always
soluble, and we can always find the final velocities of two particles in an
elastic collision. This is a powerful use of both conservation laws we have
seen so far--the two work wonderfully to predict the outcome of elastic
collisions.

Inelastic Collisions

So what if energy is not conserved? Our knowledge of such situations is more
limited, since we no longer know what the kinetic energy is after the collision.
However, even though kinetic energy is not conserved, momentum will always be
conserved. This allows us to make some statements about inelastic
collisions. Specifically, if we are given the masses of the particles, both
initial velocities and one final velocity we can calculate the final velocity of
the last particle through the familiar equation: